Why didn't the Mitzryim magic makers just take away the blood from the first makkah to prove to Moshe that he was just doing magic and Hashem had nothing to do with it: why did they add blood (Exodus 7:22)?
Imagine there was a magician claiming to have a super-natural ability to turn water into blood, and you want to discredit him and prove that it's just a trick. You would need to perform the exact illusion that the magician was performing, turn water into blood. Doing the reverse would not discredit the initial "miracle" that the magician performed.
Similarly, the Egyptian magicians wanted to show Pharaoh that what Hashem could do wasn't special. In order to do that the magicians needed to repeat the same exact effect that Moshe performed. Turning it back into water would not discredit Hashem's powerful ability to turn it to blood.
Rav Hirsch suggests an alternate explanation of the magicians’ behavior according to your suggestion: that they were attempting to undo the effects of the plague with no success—or in the case of the frogs, more frogs came when they attempted to banish them. After their third failure, they acknowledged that it was “God’s Finger” at work.
This can be explained with the interpretation of R. Yosef Bechor Shor. In his commentary to Exodus 7:20 he writes as follows:
נראה לי שלא נעשה היאור דם אך אותה שעה ובאותה שעה שנקרש ונעשה דם מתה הדגה ואחר כך נעשה מים תדע דלא אמר שלא יכלו לשתות מן היאור מפני שהוא דם כי אם מפני שמתה הדגה ונסרחה נבלתם ובאש היאור וגם אומר שעשו כן החרטומים ומה עשו הלא הכל דם אפילו בכלים כמו שהוא אומר ובעצים ובאבנים אלא ודאי לפי שעה היה דם בכל ארץ מצרים ואחר כך נהפך למים ואחר כך חזרו החרטומים והפכוהו כמו כן לדם לפי שעה ולפיכך לא אמר להם שיהפכוהו למים אלא כיון שראה שנהפך מיד למים לא שת לבו לזאת
It appears to me that the river did not become blood, except for a moment. And in that moment when it coagulated and became blood, the fish died. And afterwards it turned [back] to water. The proof is, that it does not say that they could not drink water from the river because it was blood; rather, because the fish had died and their carcasses had putrefied and polluted the river. And it also says that the magicians did the same. And what did they do? Was not everything blood, even in vessels, as it says: "in [utensils of] wood and in [utensils of] stone". Rather, certainly the blood was in all the land of Egypt for [but] a moment and then turned [back] to water, and then the magicians also turned it back into blood for a moment. And therefore, he did not say to them that they should turn [the blood back] into water; rather, once he saw that [the blood] turned immediately [back] into water, he didn't set his heart to this.
So essentially, the answer to your question is that the Egyptians could not take away the blood, because the blood had already disappeared on its own.
In fact, in his explanation of the third plague, R. Yosef Bechor Shor explicitly states that removing the plague would have been a better comeback than duplicating the plague:
נראה לי שלא רצו לעשותם אלא להסירם ולבטל מה שעשה משה כי אמרו דם שעשה משה אינו דם משה עשה צפרדעים ואינו צפרדעים הרי אנו מוסיפים מכה על מכותינו ועוד שאין זה הבחנה שמא אנו עושין במכשפות והוא על יד הקב"ה אבל בזאת נבחין שאם נוכל לבטל מה שהוא עושה אז נדע כי מכשפות הוא כי כל דבר שנעשה על ידי מכשפות מתבטל אבל [אם] לא נוכל לבטלו נדע כי אצבע אלהים היא
It appears to me that they did not want to make them, but to remove them and undo what Moses had done. For they said, "the blood that Moses made is not blood; Moses made frogs but they aren't frogs. Behold we are adding plague to our plague. And furthermore, this is not a [way to] test whether we are doing it via sorcery and he is doing it via God. But this is the [way to] test: if we are able to undo what he does then we will know that it was through sorcery, for anything done through sorcery can be undone. But if we are not able to undo it, we will know that it is the finger of God.
Thus, this further reinforces the claim that it was only because Moses's blood had reverted to water that the Egyptians chose to turn water into blood.